10x^2=10x+23

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Solution for 10x^2=10x+23 equation:


Simplifying
10x2 = 10x + 23

Reorder the terms:
10x2 = 23 + 10x

Solving
10x2 = 23 + 10x

Solving for variable 'x'.

Reorder the terms:
-23 + -10x + 10x2 = 23 + 10x + -23 + -10x

Reorder the terms:
-23 + -10x + 10x2 = 23 + -23 + 10x + -10x

Combine like terms: 23 + -23 = 0
-23 + -10x + 10x2 = 0 + 10x + -10x
-23 + -10x + 10x2 = 10x + -10x

Combine like terms: 10x + -10x = 0
-23 + -10x + 10x2 = 0

Begin completing the square.  Divide all terms by
10 the coefficient of the squared term: 

Divide each side by '10'.
-2.3 + -1x + x2 = 0

Move the constant term to the right:

Add '2.3' to each side of the equation.
-2.3 + -1x + 2.3 + x2 = 0 + 2.3

Reorder the terms:
-2.3 + 2.3 + -1x + x2 = 0 + 2.3

Combine like terms: -2.3 + 2.3 = 0.0
0.0 + -1x + x2 = 0 + 2.3
-1x + x2 = 0 + 2.3

Combine like terms: 0 + 2.3 = 2.3
-1x + x2 = 2.3

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 2.3 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 2.3 + 0.25

Combine like terms: 2.3 + 0.25 = 2.55
0.25 + -1x + x2 = 2.55

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 2.55

Calculate the square root of the right side: 1.596871942

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.596871942 and -1.596871942.

Subproblem 1

x + -0.5 = 1.596871942 Simplifying x + -0.5 = 1.596871942 Reorder the terms: -0.5 + x = 1.596871942 Solving -0.5 + x = 1.596871942 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.596871942 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.596871942 + 0.5 x = 1.596871942 + 0.5 Combine like terms: 1.596871942 + 0.5 = 2.096871942 x = 2.096871942 Simplifying x = 2.096871942

Subproblem 2

x + -0.5 = -1.596871942 Simplifying x + -0.5 = -1.596871942 Reorder the terms: -0.5 + x = -1.596871942 Solving -0.5 + x = -1.596871942 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.596871942 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.596871942 + 0.5 x = -1.596871942 + 0.5 Combine like terms: -1.596871942 + 0.5 = -1.096871942 x = -1.096871942 Simplifying x = -1.096871942

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.096871942, -1.096871942}

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